KENNELLY. — 03CILLATING-CURRENT CIRCUITS. 417 



This is a maximum when t ^= t ; when 



i^ = Q^ir^ = — e"\ amperes (94) 



T 



The power in the circuit is 



p=. l\Q^iHr^'^{U + \). watts (95). 



We may also derive (91) from the limiting case of the oscillatory 

 discharge. Taking (27), we have 



u = ^0 cosec ^ €"■'' sin {ojf + 0), volts (96) 



and if m becomes very small, as in Figure 20, the angle <^ becomes very 



small ; so that cosec <^ may be replaced by 1 /^ and sin (oj^; + 4>) by 



{cot + 0). Hence 



IT 

 u^^Q = -^ e-^t {^t + ^). volts (97) 



9 



But CO = -i-cji when </> approaches zero ; thus, 



u^^o = -^ €-^« (icfit + c^) = C^o^"'' (-"i + 1), volts (98) 

 9 



as in (91). So that the aperiodic case may be computed either as the 

 limiting oscillatory case with w = 0, or as the limiting ultraperiodic 

 case with n = 0. 



Summary of Conclusions. 



The orthogonal -projection or rotating- crank vector-diagram of the 

 ordinary a. c. circuit applies also, by extension, to the o. c. circuit. 



With the aid of the stationary vector-diagrams, the principal features 

 of any given o. c. case may be simply and speedily deduced. 



By making the above stationary vector- diagrams rotative, and subse- 

 quently applying the proper damping-factor, the process of oscillation 

 in any given o. c. circuit may be readily visualised. 



By interpreting the above diagrams and formulas hyperbolically, the 

 corresponding properties of the ultraperiodic case may be readily com- 

 puted and visualised. That is, the rotating-crank vector- diagram of 

 the ordinary a. c. circuit applies also, by extension, to the non -oscillatory 

 ultraperiodic condenser circuit. 



The properties of the condenser circuit, whether oscillatory or ultra- 



VOL. XLVI. — 27 



